P(x) has two local maxima and one local minimum. Answer the following

In summary, the polynomial P(x) has two local maxima and one local minimum, with these being the only critical points. Based on the possible graphs, the leading coefficient of P(x) would be negative.
  • #1
Painguy
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0

Homework Statement



Assume that the polynomial P(x) has exactly two local maxima and one local minimum, and that these are the only critical points of P(x). Sketch possible graphs of P(x) and use them to answer the following.
(e) What is the sign of the leading coefficient of P(x)?
positive
negative

The Attempt at a Solution



I'm not sure how to get the sign of the graph
 
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  • #2
Well think about it, your graph would go from a max, to a min, back to a max.

So by the time it was done, the graph would be heading towards negative infinity would it not?

So what does that tell you about the co-efficient of your first term?
 
  • #3
Zondrina said:
Well think about it, your graph would go from a max, to a min, back to a max.

So by the time it was done, the graph would be heading towards negative infinity would it not?

So what does that tell you about the co-efficient of your first term?

ooooo haha well that's silly of me. For some reason I had the image of a sine graph stuck in my mind. That makes perfect sense thank you.
 

1. What is a local maximum and a local minimum?

A local maximum is the highest point in a specific area of a graph, while a local minimum is the lowest point in that same area. These points represent the peak and valley of a function, respectively.

2. How can a function have two local maxima and one local minimum?

This can occur when the function has a U-shaped graph, with one side having a higher peak than the other. The highest point on each side would be a local maximum, while the bottom of the U-shape would be the local minimum.

3. What does it mean for a function to have multiple local maxima and a single local minimum?

When a function has multiple local maxima, it means that there are multiple peaks in the graph. The single local minimum represents the lowest point in the graph. This type of function can have a variety of shapes, such as a W-shape or a series of hills and valleys.

4. How can you find the coordinates of the local maxima and local minimum in a function?

To find the coordinates of the local maxima and local minimum, you can use the first and second derivative tests. The first derivative test involves finding the critical points of the function, where the derivative is equal to zero. The second derivative test then determines whether these critical points are local maxima or local minima.

5. What does the presence of local maxima and local minimum indicate about a function?

The presence of local maxima and local minimum indicates that the function is not a straight line and has varying rates of change. It also shows that the function has both increasing and decreasing intervals, where the slope changes from positive to negative or vice versa. This can provide valuable insights into the behavior of the function and its relationship with its input variables.

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